Non-commutativity effects in the Dirac equation in crossed electric and magnetic fields

¹ Department of Mathematics, Vivekananda College - Kolkata-700063, India
² Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova - Padova, Italy
³ Istituto Nazionale di Fisica Nucleare, Sezione di Padova - Padova, Italy
⁴ Istituto Nazionale di Fisica Nucleare, Sezione di Perugia - Via A. Pascoli, Perugia, Italy
⁵ Physics and Applied Mathematics Unit, Indian Statistical Institute - Kolkata-700108, India

Received: 7 May 2018
Accepted: 26 July 2018

Abstract

In this paper we present exact solutions of the Dirac equation on the non-commutative plane in the presence of crossed electric and magnetic fields. In the standard commutative plane such a system is known to exhibit contraction of Landau levels when the electric field approaches a critical value. In the present case we find exact solutions in terms of the non-commutative parameters η (momentum non-commutativity) and θ (coordinate non-commutativity) and provide an explicit expression for the Landau levels. We show that non-commutativity preserves the collapse of the spectrum. We provide a dual description of the system: i) one in which at a given electric field the magnetic field is varied and the other ii) in which at a given magnetic field the electric field is varied. In the former case we find that momentum non-commutativity (η) splits the critical magnetic field into two critical fields while coordinates non-commutativity (θ) gives rise to two additional critical points not at all present in the commutative scenario.